Second answer: do you want to have a controlled environment for your testing?

Third answer: could failure in test server affect customer service? For example by delaying a "live" change.

Fourth answer: could the test and live servers being out of step lead to a problem in the live server with something that was tested on a differently patched test server?

Fifth answer: if you are using the test server to test the patch does other test work need to be suspended pending completion or regression?_________________"Method goes far to prevent trouble in business: for it makes the task easy, hinders confusion, saves abundance of time, and instructs those that have business depending, both what to do and what to hope."
William Penn 1644-1718

What do your procedures and policies say?
Till now nothing has mentioned in procudure for the changes to test environment. Thats why only the question has come in picture now.

Second answer: do you want to have a controlled environment for your testing?
This is always and everywhere desirable.

Third answer: could failure in test server affect customer service? For example by delaying a "live" change.
depends on the consitionj, some times it can affect some time it wont.

Fourth answer: could the test and live servers being out of step lead to a problem in the live server with something that was tested on a differently patched test server?
We can not deny the possibility of such case.

the answer of All the questions you have puted on will be according to type of work/change we are doing with test enviroment. Some times , It may impact production by some way and some time not linkage with Live envirement.

In such scenerio, should we bring every change in test enviroment to the change management process????

I would suggest in that case you need to fully control all such changes._________________"Method goes far to prevent trouble in business: for it makes the task easy, hinders confusion, saves abundance of time, and instructs those that have business depending, both what to do and what to hope."
William Penn 1644-1718